SOLUTION: The mean useful life of car batteries is 55 months. They have a standard deviation of 2. Assume the useful life of batteries is normally distributed. a. Calculate the percent

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Question 1190332: The mean useful life of car batteries is 55 months. They have a standard deviation of 2. Assume the useful life of batteries is normally distributed.

a. Calculate the percent of batteries with a useful life of less than 51 months. (Round your answer to the nearest hundredth percent.)
b. Calculate the percent of batteries that will last longer than 61 months. (Round your answer to the nearest hundredth percent.)

Answer by Theo(13342) (Show Source):
You can put this solution on YOUR website!
mean = 55
standard deviation = 2
z-score = (x - m) / s
x = the raw score = 51
m = the mean = 55
s = standard deviation = 2

for part a ....

z = (x - m) / s becomes z = (51 - 55) / 2 = -4 / 2 = -2
area to the left of z-score of -2 = .02275
rounded to 4 decimal places to bet .0228 = 2.28%.

for part b ....

z = (x - m) / s becomes z = (61 - 55) / 2 = 6/2 = 3
area to the left of z-score of 3 = .99865.
area to the right of z-score of 3 = 1 - .99865 = .00135.
round to 4 decimal places to get .0014 = .14%.

i used the following z-score table to get these figures.

https://www.rit.edu/academicsuccesscenter/sites/rit.edu.academicsuccesscenter/files/documents/math-handouts/Standard%20Normal%20Distribution%20Table.pdf

using the ti-85 plus calculator, i got:

area to the left of z-score of -2 = .022750062.
rounded to 4 decimal places = .0228 = 2.28%.

area to the left of z-score of 3 = .9986500328.
area to the right of z-score of 3 = 1 minus .9986500328 = .0013499672.
rounded to 4 decimal places = .0013 = .13%.

the discrepancy between .13% and .14% has to do with the table figures being rounded to 5 decimal places while the calculator figures are being rounded to 7 or 8 decimal places.
the difference affected the rounding to 4 decimal places in this case.
it's a small difference, but it is a difference.
if you go by the table, then .14% is more accurate.